关于具有非负电阻曲率的图的一些结构性质
On some structural properties of graphs with non-negative resistance curvature
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中文总结 AI 辅助
研究具有非负电阻曲率的图的结构，通过构造反例反驳Fiedler猜想、否定回答Devriendt问题，证明Thomassen 34 - 图性质及路径笛卡尔积是RN图，解决了关于韧性、可迹性及网格图的相关猜想。
中文摘要 AI 辅助
如果一个图允许正边权使得所有顶点电阻曲率是非负的（分别为正的），则称该图为电阻非负（RN）图，分别为电阻正（RP）图。本文研究了RN和RP图在韧性、可迹性和笛卡尔积方面的结构。首先，通过构造每个\(n\geq11\)的\(n\)顶点\(1 -\)韧性图且不是RN图，反驳了Fiedler的一个猜想并否定回答了Devriendt的一个问题。其次，通过证明Thomassen 34 - 图是RP但不可迹，表明RP图不一定可迹。最后，通过证明路径的所有笛卡尔积都是RN，解决了Devriendt关于网格图的一个猜想。
英文摘要
A graph is called resistance nonnegative (RN), respectively resistance positive (RP), if it admits positive edge weights such that all vertex resistance curvatures are nonnegative, respectively positive. In this paper, we study the structure of RN and RP graphs in relation to toughness, traceability, and Cartesian products. First, we disprove a conjecture of Fiedler and answer a question of Devriendt in the negative by constructing, for every $n\ge 11$, an $n$-vertex $1$-tough graph that is not RN. Second, we show that RP graphs need not be traceable by proving that the Thomassen $34$-graph is RP but not traceable. Finally, we resolve a conjecture of Devriendt on grid graphs by proving that all Cartesian products of paths are RN.
